Theorem dilsetN 40352

Description: The set of dilations for a fiducial atom 𝐷. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dilset.a	⊢ 𝐴 = (Atoms‘𝐾)
dilset.s	⊢ 𝑆 = (PSubSp‘𝐾)
dilset.w	⊢ 𝑊 = (WAtoms‘𝐾)
dilset.m	⊢ 𝑀 = (PAut‘𝐾)
dilset.l	⊢ 𝐿 = (Dil‘𝐾)

Assertion

Ref	Expression
dilsetN	⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐿‘𝐷) = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)})

Distinct variable groups:   𝑥,𝑓,𝐾   𝑓,𝑀   𝑥,𝑆   𝐷,𝑓,𝑥

Allowed substitution hints:   𝐴(𝑥,𝑓)   𝐵(𝑥,𝑓)   𝑆(𝑓)   𝐿(𝑥,𝑓)   𝑀(𝑥)   𝑊(𝑥,𝑓)

Proof of Theorem dilsetN

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dilset.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
2		dilset.s	. . . 4 ⊢ 𝑆 = (PSubSp‘𝐾)
3		dilset.w	. . . 4 ⊢ 𝑊 = (WAtoms‘𝐾)
4		dilset.m	. . . 4 ⊢ 𝑀 = (PAut‘𝐾)
5		dilset.l	. . . 4 ⊢ 𝐿 = (Dil‘𝐾)
6	1, 2, 3, 4, 5	dilfsetN 40351	. . 3 ⊢ (𝐾 ∈ 𝐵 → 𝐿 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}))
7	6	fveq1d 6834	. 2 ⊢ (𝐾 ∈ 𝐵 → (𝐿‘𝐷) = ((𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)})‘𝐷))
8		fveq2 6832	. . . . . . 7 ⊢ (𝑑 = 𝐷 → (𝑊‘𝑑) = (𝑊‘𝐷))
9	8	sseq2d 3964	. . . . . 6 ⊢ (𝑑 = 𝐷 → (𝑥 ⊆ (𝑊‘𝑑) ↔ 𝑥 ⊆ (𝑊‘𝐷)))
10	9	imbi1d 341	. . . . 5 ⊢ (𝑑 = 𝐷 → ((𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥) ↔ (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)))
11	10	ralbidv 3157	. . . 4 ⊢ (𝑑 = 𝐷 → (∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥) ↔ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)))
12	11	rabbidv 3404	. . 3 ⊢ (𝑑 = 𝐷 → {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)} = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)})
13		eqid 2734	. . 3 ⊢ (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)}) = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)})
14	4	fvexi 6846	. . . 4 ⊢ 𝑀 ∈ V
15	14	rabex 5282	. . 3 ⊢ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)} ∈ V
16	12, 13, 15	fvmpt 6939	. 2 ⊢ (𝐷 ∈ 𝐴 → ((𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝑑) → (𝑓‘𝑥) = 𝑥)})‘𝐷) = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)})
17	7, 16	sylan9eq 2789	1 ⊢ ((𝐾 ∈ 𝐵 ∧ 𝐷 ∈ 𝐴) → (𝐿‘𝐷) = {𝑓 ∈ 𝑀 ∣ ∀𝑥 ∈ 𝑆 (𝑥 ⊆ (𝑊‘𝐷) → (𝑓‘𝑥) = 𝑥)})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3049  {crab 3397   ⊆ wss 3899   ↦ cmpt 5177  ‘cfv 6490  Atomscatm 39462  PSubSpcpsubsp 39695  WAtomscwpointsN 40185  PAutcpautN 40186  DilcdilN 40301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-dilN 40305

This theorem is referenced by:  isdilN  40353